MAYBE

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X) -> U11(X)
  , a__U11(tt()) -> tt()
  , a__U21(X) -> U21(X)
  , a__U21(tt()) -> tt()
  , a__U31(X) -> U31(X)
  , a__U31(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), V2) -> a__U42(a__isNatIList(V2))
  , a__U42(X) -> U42(X)
  , a__U42(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatList(V))
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), V2) -> a__U52(a__isNatList(V2))
  , a__U52(X) -> U52(X)
  , a__U52(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2)
  , a__isNatList(nil()) -> tt()
  , a__isNatList(take(V1, V2)) -> a__U61(a__isNat(V1), V2)
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), V2) -> a__U62(a__isNatIList(V2))
  , a__U62(X) -> U62(X)
  , a__U62(tt()) -> tt()
  , a__U71(X1, X2, X3) -> U71(X1, X2, X3)
  , a__U71(tt(), L, N) -> a__U72(a__isNat(N), L)
  , a__U72(X1, X2) -> U72(X1, X2)
  , a__U72(tt(), L) -> s(a__length(mark(L)))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNat(V1))
  , a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) -> a__U71(a__isNatList(L), L, N)
  , a__length(nil()) -> 0()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(0()) -> 0()
  , mark(zeros()) -> a__zeros()
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(nil()) -> nil()
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(length(X)) -> a__length(mark(X))
  , mark(U11(X)) -> a__U11(mark(X))
  , mark(U21(X)) -> a__U21(mark(X))
  , mark(U31(X)) -> a__U31(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U42(X)) -> a__U42(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X)) -> a__U52(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , mark(U62(X)) -> a__U62(mark(X))
  , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
  , mark(U72(X1, X2)) -> a__U72(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U81(X)) -> a__U81(mark(X))
  , mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4)
  , mark(U92(X1, X2, X3, X4)) -> a__U92(mark(X1), X2, X3, X4)
  , mark(U93(X1, X2, X3, X4)) -> a__U93(mark(X1), X2, X3, X4)
  , a__U81(X) -> U81(X)
  , a__U81(tt()) -> nil()
  , a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4)
  , a__U91(tt(), IL, M, N) -> a__U92(a__isNat(M), IL, M, N)
  , a__U92(X1, X2, X3, X4) -> U92(X1, X2, X3, X4)
  , a__U92(tt(), IL, M, N) -> a__U93(a__isNat(N), IL, M, N)
  , a__U93(X1, X2, X3, X4) -> U93(X1, X2, X3, X4)
  , a__U93(tt(), IL, M, N) -> cons(mark(N), take(M, IL))
  , a__take(X1, X2) -> take(X1, X2)
  , a__take(0(), IL) -> a__U81(a__isNatIList(IL))
  , a__take(s(M), cons(N, IL)) ->
    a__U91(a__isNatIList(IL), IL, M, N) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { a__zeros^#() -> c_1()
  , a__zeros^#() -> c_2()
  , a__U11^#(X) -> c_3()
  , a__U11^#(tt()) -> c_4()
  , a__U21^#(X) -> c_5()
  , a__U21^#(tt()) -> c_6()
  , a__U31^#(X) -> c_7()
  , a__U31^#(tt()) -> c_8()
  , a__U41^#(X1, X2) -> c_9()
  , a__U41^#(tt(), V2) ->
    c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__U42^#(X) -> c_11()
  , a__U42^#(tt()) -> c_12()
  , a__isNatIList^#(V) ->
    c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
  , a__isNatIList^#(X) -> c_14()
  , a__isNatIList^#(cons(V1, V2)) ->
    c_15(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatIList^#(zeros()) -> c_16()
  , a__isNatList^#(X) -> c_21()
  , a__isNatList^#(cons(V1, V2)) ->
    c_22(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(nil()) -> c_23()
  , a__isNatList^#(take(V1, V2)) ->
    c_24(a__U61^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNat^#(X) -> c_33()
  , a__isNat^#(0()) -> c_34()
  , a__isNat^#(s(V1)) -> c_35(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(length(V1)) ->
    c_36(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U51^#(X1, X2) -> c_17()
  , a__U51^#(tt(), V2) ->
    c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U52^#(X) -> c_19()
  , a__U52^#(tt()) -> c_20()
  , a__U61^#(X1, X2) -> c_25()
  , a__U61^#(tt(), V2) ->
    c_26(a__U62^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__U62^#(X) -> c_27()
  , a__U62^#(tt()) -> c_28()
  , a__U71^#(X1, X2, X3) -> c_29()
  , a__U71^#(tt(), L, N) ->
    c_30(a__U72^#(a__isNat(N), L), a__isNat^#(N))
  , a__U72^#(X1, X2) -> c_31()
  , a__U72^#(tt(), L) -> c_32(a__length^#(mark(L)), mark^#(L))
  , a__length^#(X) -> c_37()
  , a__length^#(cons(N, L)) ->
    c_38(a__U71^#(a__isNatList(L), L, N), a__isNatList^#(L))
  , a__length^#(nil()) -> c_39()
  , mark^#(cons(X1, X2)) -> c_40(mark^#(X1))
  , mark^#(0()) -> c_41()
  , mark^#(zeros()) -> c_42(a__zeros^#())
  , mark^#(tt()) -> c_43()
  , mark^#(s(X)) -> c_44(mark^#(X))
  , mark^#(nil()) -> c_45()
  , mark^#(take(X1, X2)) ->
    c_46(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(length(X)) -> c_47(a__length^#(mark(X)), mark^#(X))
  , mark^#(U11(X)) -> c_48(a__U11^#(mark(X)), mark^#(X))
  , mark^#(U21(X)) -> c_49(a__U21^#(mark(X)), mark^#(X))
  , mark^#(U31(X)) -> c_50(a__U31^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2)) -> c_51(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U42(X)) -> c_52(a__U42^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_53(a__isNatIList^#(X))
  , mark^#(U51(X1, X2)) -> c_54(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X)) -> c_55(a__U52^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_56(a__isNatList^#(X))
  , mark^#(U61(X1, X2)) -> c_57(a__U61^#(mark(X1), X2), mark^#(X1))
  , mark^#(U62(X)) -> c_58(a__U62^#(mark(X)), mark^#(X))
  , mark^#(U71(X1, X2, X3)) ->
    c_59(a__U71^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U72(X1, X2)) -> c_60(a__U72^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_61(a__isNat^#(X))
  , mark^#(U81(X)) -> c_62(a__U81^#(mark(X)), mark^#(X))
  , mark^#(U91(X1, X2, X3, X4)) ->
    c_63(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U92(X1, X2, X3, X4)) ->
    c_64(a__U92^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U93(X1, X2, X3, X4)) ->
    c_65(a__U93^#(mark(X1), X2, X3, X4), mark^#(X1))
  , a__take^#(X1, X2) -> c_74()
  , a__take^#(0(), IL) ->
    c_75(a__U81^#(a__isNatIList(IL)), a__isNatIList^#(IL))
  , a__take^#(s(M), cons(N, IL)) ->
    c_76(a__U91^#(a__isNatIList(IL), IL, M, N), a__isNatIList^#(IL))
  , a__U81^#(X) -> c_66()
  , a__U81^#(tt()) -> c_67()
  , a__U91^#(X1, X2, X3, X4) -> c_68()
  , a__U91^#(tt(), IL, M, N) ->
    c_69(a__U92^#(a__isNat(M), IL, M, N), a__isNat^#(M))
  , a__U92^#(X1, X2, X3, X4) -> c_70()
  , a__U92^#(tt(), IL, M, N) ->
    c_71(a__U93^#(a__isNat(N), IL, M, N), a__isNat^#(N))
  , a__U93^#(X1, X2, X3, X4) -> c_72()
  , a__U93^#(tt(), IL, M, N) -> c_73(mark^#(N)) }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { a__zeros^#() -> c_1()
  , a__zeros^#() -> c_2()
  , a__U11^#(X) -> c_3()
  , a__U11^#(tt()) -> c_4()
  , a__U21^#(X) -> c_5()
  , a__U21^#(tt()) -> c_6()
  , a__U31^#(X) -> c_7()
  , a__U31^#(tt()) -> c_8()
  , a__U41^#(X1, X2) -> c_9()
  , a__U41^#(tt(), V2) ->
    c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__U42^#(X) -> c_11()
  , a__U42^#(tt()) -> c_12()
  , a__isNatIList^#(V) ->
    c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
  , a__isNatIList^#(X) -> c_14()
  , a__isNatIList^#(cons(V1, V2)) ->
    c_15(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatIList^#(zeros()) -> c_16()
  , a__isNatList^#(X) -> c_21()
  , a__isNatList^#(cons(V1, V2)) ->
    c_22(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(nil()) -> c_23()
  , a__isNatList^#(take(V1, V2)) ->
    c_24(a__U61^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNat^#(X) -> c_33()
  , a__isNat^#(0()) -> c_34()
  , a__isNat^#(s(V1)) -> c_35(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(length(V1)) ->
    c_36(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U51^#(X1, X2) -> c_17()
  , a__U51^#(tt(), V2) ->
    c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U52^#(X) -> c_19()
  , a__U52^#(tt()) -> c_20()
  , a__U61^#(X1, X2) -> c_25()
  , a__U61^#(tt(), V2) ->
    c_26(a__U62^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__U62^#(X) -> c_27()
  , a__U62^#(tt()) -> c_28()
  , a__U71^#(X1, X2, X3) -> c_29()
  , a__U71^#(tt(), L, N) ->
    c_30(a__U72^#(a__isNat(N), L), a__isNat^#(N))
  , a__U72^#(X1, X2) -> c_31()
  , a__U72^#(tt(), L) -> c_32(a__length^#(mark(L)), mark^#(L))
  , a__length^#(X) -> c_37()
  , a__length^#(cons(N, L)) ->
    c_38(a__U71^#(a__isNatList(L), L, N), a__isNatList^#(L))
  , a__length^#(nil()) -> c_39()
  , mark^#(cons(X1, X2)) -> c_40(mark^#(X1))
  , mark^#(0()) -> c_41()
  , mark^#(zeros()) -> c_42(a__zeros^#())
  , mark^#(tt()) -> c_43()
  , mark^#(s(X)) -> c_44(mark^#(X))
  , mark^#(nil()) -> c_45()
  , mark^#(take(X1, X2)) ->
    c_46(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(length(X)) -> c_47(a__length^#(mark(X)), mark^#(X))
  , mark^#(U11(X)) -> c_48(a__U11^#(mark(X)), mark^#(X))
  , mark^#(U21(X)) -> c_49(a__U21^#(mark(X)), mark^#(X))
  , mark^#(U31(X)) -> c_50(a__U31^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2)) -> c_51(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U42(X)) -> c_52(a__U42^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_53(a__isNatIList^#(X))
  , mark^#(U51(X1, X2)) -> c_54(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X)) -> c_55(a__U52^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_56(a__isNatList^#(X))
  , mark^#(U61(X1, X2)) -> c_57(a__U61^#(mark(X1), X2), mark^#(X1))
  , mark^#(U62(X)) -> c_58(a__U62^#(mark(X)), mark^#(X))
  , mark^#(U71(X1, X2, X3)) ->
    c_59(a__U71^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U72(X1, X2)) -> c_60(a__U72^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_61(a__isNat^#(X))
  , mark^#(U81(X)) -> c_62(a__U81^#(mark(X)), mark^#(X))
  , mark^#(U91(X1, X2, X3, X4)) ->
    c_63(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U92(X1, X2, X3, X4)) ->
    c_64(a__U92^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U93(X1, X2, X3, X4)) ->
    c_65(a__U93^#(mark(X1), X2, X3, X4), mark^#(X1))
  , a__take^#(X1, X2) -> c_74()
  , a__take^#(0(), IL) ->
    c_75(a__U81^#(a__isNatIList(IL)), a__isNatIList^#(IL))
  , a__take^#(s(M), cons(N, IL)) ->
    c_76(a__U91^#(a__isNatIList(IL), IL, M, N), a__isNatIList^#(IL))
  , a__U81^#(X) -> c_66()
  , a__U81^#(tt()) -> c_67()
  , a__U91^#(X1, X2, X3, X4) -> c_68()
  , a__U91^#(tt(), IL, M, N) ->
    c_69(a__U92^#(a__isNat(M), IL, M, N), a__isNat^#(M))
  , a__U92^#(X1, X2, X3, X4) -> c_70()
  , a__U92^#(tt(), IL, M, N) ->
    c_71(a__U93^#(a__isNat(N), IL, M, N), a__isNat^#(N))
  , a__U93^#(X1, X2, X3, X4) -> c_72()
  , a__U93^#(tt(), IL, M, N) -> c_73(mark^#(N)) }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X) -> U11(X)
  , a__U11(tt()) -> tt()
  , a__U21(X) -> U21(X)
  , a__U21(tt()) -> tt()
  , a__U31(X) -> U31(X)
  , a__U31(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), V2) -> a__U42(a__isNatIList(V2))
  , a__U42(X) -> U42(X)
  , a__U42(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatList(V))
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), V2) -> a__U52(a__isNatList(V2))
  , a__U52(X) -> U52(X)
  , a__U52(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2)
  , a__isNatList(nil()) -> tt()
  , a__isNatList(take(V1, V2)) -> a__U61(a__isNat(V1), V2)
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), V2) -> a__U62(a__isNatIList(V2))
  , a__U62(X) -> U62(X)
  , a__U62(tt()) -> tt()
  , a__U71(X1, X2, X3) -> U71(X1, X2, X3)
  , a__U71(tt(), L, N) -> a__U72(a__isNat(N), L)
  , a__U72(X1, X2) -> U72(X1, X2)
  , a__U72(tt(), L) -> s(a__length(mark(L)))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNat(V1))
  , a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) -> a__U71(a__isNatList(L), L, N)
  , a__length(nil()) -> 0()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(0()) -> 0()
  , mark(zeros()) -> a__zeros()
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(nil()) -> nil()
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(length(X)) -> a__length(mark(X))
  , mark(U11(X)) -> a__U11(mark(X))
  , mark(U21(X)) -> a__U21(mark(X))
  , mark(U31(X)) -> a__U31(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U42(X)) -> a__U42(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X)) -> a__U52(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , mark(U62(X)) -> a__U62(mark(X))
  , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
  , mark(U72(X1, X2)) -> a__U72(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U81(X)) -> a__U81(mark(X))
  , mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4)
  , mark(U92(X1, X2, X3, X4)) -> a__U92(mark(X1), X2, X3, X4)
  , mark(U93(X1, X2, X3, X4)) -> a__U93(mark(X1), X2, X3, X4)
  , a__U81(X) -> U81(X)
  , a__U81(tt()) -> nil()
  , a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4)
  , a__U91(tt(), IL, M, N) -> a__U92(a__isNat(M), IL, M, N)
  , a__U92(X1, X2, X3, X4) -> U92(X1, X2, X3, X4)
  , a__U92(tt(), IL, M, N) -> a__U93(a__isNat(N), IL, M, N)
  , a__U93(X1, X2, X3, X4) -> U93(X1, X2, X3, X4)
  , a__U93(tt(), IL, M, N) -> cons(mark(N), take(M, IL))
  , a__take(X1, X2) -> take(X1, X2)
  , a__take(0(), IL) -> a__U81(a__isNatIList(IL))
  , a__take(s(M), cons(N, IL)) ->
    a__U91(a__isNatIList(IL), IL, M, N) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of
{1,2,3,4,5,6,7,8,9,11,12,14,16,17,19,21,22,25,27,28,29,31,32,33,35,37,39,41,43,45,66,69,70,71,73,75}
by applications of
Pre({1,2,3,4,5,6,7,8,9,11,12,14,16,17,19,21,22,25,27,28,29,31,32,33,35,37,39,41,43,45,66,69,70,71,73,75})
=
{10,13,15,18,20,23,24,26,30,34,36,38,40,42,44,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,67,68,72,74,76}.
Here rules are labeled as follows:

  DPs:
    { 1: a__zeros^#() -> c_1()
    , 2: a__zeros^#() -> c_2()
    , 3: a__U11^#(X) -> c_3()
    , 4: a__U11^#(tt()) -> c_4()
    , 5: a__U21^#(X) -> c_5()
    , 6: a__U21^#(tt()) -> c_6()
    , 7: a__U31^#(X) -> c_7()
    , 8: a__U31^#(tt()) -> c_8()
    , 9: a__U41^#(X1, X2) -> c_9()
    , 10: a__U41^#(tt(), V2) ->
          c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
    , 11: a__U42^#(X) -> c_11()
    , 12: a__U42^#(tt()) -> c_12()
    , 13: a__isNatIList^#(V) ->
          c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
    , 14: a__isNatIList^#(X) -> c_14()
    , 15: a__isNatIList^#(cons(V1, V2)) ->
          c_15(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 16: a__isNatIList^#(zeros()) -> c_16()
    , 17: a__isNatList^#(X) -> c_21()
    , 18: a__isNatList^#(cons(V1, V2)) ->
          c_22(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 19: a__isNatList^#(nil()) -> c_23()
    , 20: a__isNatList^#(take(V1, V2)) ->
          c_24(a__U61^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 21: a__isNat^#(X) -> c_33()
    , 22: a__isNat^#(0()) -> c_34()
    , 23: a__isNat^#(s(V1)) ->
          c_35(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
    , 24: a__isNat^#(length(V1)) ->
          c_36(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
    , 25: a__U51^#(X1, X2) -> c_17()
    , 26: a__U51^#(tt(), V2) ->
          c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
    , 27: a__U52^#(X) -> c_19()
    , 28: a__U52^#(tt()) -> c_20()
    , 29: a__U61^#(X1, X2) -> c_25()
    , 30: a__U61^#(tt(), V2) ->
          c_26(a__U62^#(a__isNatIList(V2)), a__isNatIList^#(V2))
    , 31: a__U62^#(X) -> c_27()
    , 32: a__U62^#(tt()) -> c_28()
    , 33: a__U71^#(X1, X2, X3) -> c_29()
    , 34: a__U71^#(tt(), L, N) ->
          c_30(a__U72^#(a__isNat(N), L), a__isNat^#(N))
    , 35: a__U72^#(X1, X2) -> c_31()
    , 36: a__U72^#(tt(), L) -> c_32(a__length^#(mark(L)), mark^#(L))
    , 37: a__length^#(X) -> c_37()
    , 38: a__length^#(cons(N, L)) ->
          c_38(a__U71^#(a__isNatList(L), L, N), a__isNatList^#(L))
    , 39: a__length^#(nil()) -> c_39()
    , 40: mark^#(cons(X1, X2)) -> c_40(mark^#(X1))
    , 41: mark^#(0()) -> c_41()
    , 42: mark^#(zeros()) -> c_42(a__zeros^#())
    , 43: mark^#(tt()) -> c_43()
    , 44: mark^#(s(X)) -> c_44(mark^#(X))
    , 45: mark^#(nil()) -> c_45()
    , 46: mark^#(take(X1, X2)) ->
          c_46(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 47: mark^#(length(X)) -> c_47(a__length^#(mark(X)), mark^#(X))
    , 48: mark^#(U11(X)) -> c_48(a__U11^#(mark(X)), mark^#(X))
    , 49: mark^#(U21(X)) -> c_49(a__U21^#(mark(X)), mark^#(X))
    , 50: mark^#(U31(X)) -> c_50(a__U31^#(mark(X)), mark^#(X))
    , 51: mark^#(U41(X1, X2)) ->
          c_51(a__U41^#(mark(X1), X2), mark^#(X1))
    , 52: mark^#(U42(X)) -> c_52(a__U42^#(mark(X)), mark^#(X))
    , 53: mark^#(isNatIList(X)) -> c_53(a__isNatIList^#(X))
    , 54: mark^#(U51(X1, X2)) ->
          c_54(a__U51^#(mark(X1), X2), mark^#(X1))
    , 55: mark^#(U52(X)) -> c_55(a__U52^#(mark(X)), mark^#(X))
    , 56: mark^#(isNatList(X)) -> c_56(a__isNatList^#(X))
    , 57: mark^#(U61(X1, X2)) ->
          c_57(a__U61^#(mark(X1), X2), mark^#(X1))
    , 58: mark^#(U62(X)) -> c_58(a__U62^#(mark(X)), mark^#(X))
    , 59: mark^#(U71(X1, X2, X3)) ->
          c_59(a__U71^#(mark(X1), X2, X3), mark^#(X1))
    , 60: mark^#(U72(X1, X2)) ->
          c_60(a__U72^#(mark(X1), X2), mark^#(X1))
    , 61: mark^#(isNat(X)) -> c_61(a__isNat^#(X))
    , 62: mark^#(U81(X)) -> c_62(a__U81^#(mark(X)), mark^#(X))
    , 63: mark^#(U91(X1, X2, X3, X4)) ->
          c_63(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1))
    , 64: mark^#(U92(X1, X2, X3, X4)) ->
          c_64(a__U92^#(mark(X1), X2, X3, X4), mark^#(X1))
    , 65: mark^#(U93(X1, X2, X3, X4)) ->
          c_65(a__U93^#(mark(X1), X2, X3, X4), mark^#(X1))
    , 66: a__take^#(X1, X2) -> c_74()
    , 67: a__take^#(0(), IL) ->
          c_75(a__U81^#(a__isNatIList(IL)), a__isNatIList^#(IL))
    , 68: a__take^#(s(M), cons(N, IL)) ->
          c_76(a__U91^#(a__isNatIList(IL), IL, M, N), a__isNatIList^#(IL))
    , 69: a__U81^#(X) -> c_66()
    , 70: a__U81^#(tt()) -> c_67()
    , 71: a__U91^#(X1, X2, X3, X4) -> c_68()
    , 72: a__U91^#(tt(), IL, M, N) ->
          c_69(a__U92^#(a__isNat(M), IL, M, N), a__isNat^#(M))
    , 73: a__U92^#(X1, X2, X3, X4) -> c_70()
    , 74: a__U92^#(tt(), IL, M, N) ->
          c_71(a__U93^#(a__isNat(N), IL, M, N), a__isNat^#(N))
    , 75: a__U93^#(X1, X2, X3, X4) -> c_72()
    , 76: a__U93^#(tt(), IL, M, N) -> c_73(mark^#(N)) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { a__U41^#(tt(), V2) ->
    c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__isNatIList^#(V) ->
    c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
  , a__isNatIList^#(cons(V1, V2)) ->
    c_15(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(cons(V1, V2)) ->
    c_22(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(take(V1, V2)) ->
    c_24(a__U61^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNat^#(s(V1)) -> c_35(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(length(V1)) ->
    c_36(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U51^#(tt(), V2) ->
    c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U61^#(tt(), V2) ->
    c_26(a__U62^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__U71^#(tt(), L, N) ->
    c_30(a__U72^#(a__isNat(N), L), a__isNat^#(N))
  , a__U72^#(tt(), L) -> c_32(a__length^#(mark(L)), mark^#(L))
  , a__length^#(cons(N, L)) ->
    c_38(a__U71^#(a__isNatList(L), L, N), a__isNatList^#(L))
  , mark^#(cons(X1, X2)) -> c_40(mark^#(X1))
  , mark^#(zeros()) -> c_42(a__zeros^#())
  , mark^#(s(X)) -> c_44(mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_46(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(length(X)) -> c_47(a__length^#(mark(X)), mark^#(X))
  , mark^#(U11(X)) -> c_48(a__U11^#(mark(X)), mark^#(X))
  , mark^#(U21(X)) -> c_49(a__U21^#(mark(X)), mark^#(X))
  , mark^#(U31(X)) -> c_50(a__U31^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2)) -> c_51(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U42(X)) -> c_52(a__U42^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_53(a__isNatIList^#(X))
  , mark^#(U51(X1, X2)) -> c_54(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X)) -> c_55(a__U52^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_56(a__isNatList^#(X))
  , mark^#(U61(X1, X2)) -> c_57(a__U61^#(mark(X1), X2), mark^#(X1))
  , mark^#(U62(X)) -> c_58(a__U62^#(mark(X)), mark^#(X))
  , mark^#(U71(X1, X2, X3)) ->
    c_59(a__U71^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U72(X1, X2)) -> c_60(a__U72^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_61(a__isNat^#(X))
  , mark^#(U81(X)) -> c_62(a__U81^#(mark(X)), mark^#(X))
  , mark^#(U91(X1, X2, X3, X4)) ->
    c_63(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U92(X1, X2, X3, X4)) ->
    c_64(a__U92^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U93(X1, X2, X3, X4)) ->
    c_65(a__U93^#(mark(X1), X2, X3, X4), mark^#(X1))
  , a__take^#(0(), IL) ->
    c_75(a__U81^#(a__isNatIList(IL)), a__isNatIList^#(IL))
  , a__take^#(s(M), cons(N, IL)) ->
    c_76(a__U91^#(a__isNatIList(IL), IL, M, N), a__isNatIList^#(IL))
  , a__U91^#(tt(), IL, M, N) ->
    c_69(a__U92^#(a__isNat(M), IL, M, N), a__isNat^#(M))
  , a__U92^#(tt(), IL, M, N) ->
    c_71(a__U93^#(a__isNat(N), IL, M, N), a__isNat^#(N))
  , a__U93^#(tt(), IL, M, N) -> c_73(mark^#(N)) }
Weak DPs:
  { a__zeros^#() -> c_1()
  , a__zeros^#() -> c_2()
  , a__U11^#(X) -> c_3()
  , a__U11^#(tt()) -> c_4()
  , a__U21^#(X) -> c_5()
  , a__U21^#(tt()) -> c_6()
  , a__U31^#(X) -> c_7()
  , a__U31^#(tt()) -> c_8()
  , a__U41^#(X1, X2) -> c_9()
  , a__U42^#(X) -> c_11()
  , a__U42^#(tt()) -> c_12()
  , a__isNatIList^#(X) -> c_14()
  , a__isNatIList^#(zeros()) -> c_16()
  , a__isNatList^#(X) -> c_21()
  , a__isNatList^#(nil()) -> c_23()
  , a__isNat^#(X) -> c_33()
  , a__isNat^#(0()) -> c_34()
  , a__U51^#(X1, X2) -> c_17()
  , a__U52^#(X) -> c_19()
  , a__U52^#(tt()) -> c_20()
  , a__U61^#(X1, X2) -> c_25()
  , a__U62^#(X) -> c_27()
  , a__U62^#(tt()) -> c_28()
  , a__U71^#(X1, X2, X3) -> c_29()
  , a__U72^#(X1, X2) -> c_31()
  , a__length^#(X) -> c_37()
  , a__length^#(nil()) -> c_39()
  , mark^#(0()) -> c_41()
  , mark^#(tt()) -> c_43()
  , mark^#(nil()) -> c_45()
  , a__take^#(X1, X2) -> c_74()
  , a__U81^#(X) -> c_66()
  , a__U81^#(tt()) -> c_67()
  , a__U91^#(X1, X2, X3, X4) -> c_68()
  , a__U92^#(X1, X2, X3, X4) -> c_70()
  , a__U93^#(X1, X2, X3, X4) -> c_72() }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X) -> U11(X)
  , a__U11(tt()) -> tt()
  , a__U21(X) -> U21(X)
  , a__U21(tt()) -> tt()
  , a__U31(X) -> U31(X)
  , a__U31(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), V2) -> a__U42(a__isNatIList(V2))
  , a__U42(X) -> U42(X)
  , a__U42(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatList(V))
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), V2) -> a__U52(a__isNatList(V2))
  , a__U52(X) -> U52(X)
  , a__U52(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2)
  , a__isNatList(nil()) -> tt()
  , a__isNatList(take(V1, V2)) -> a__U61(a__isNat(V1), V2)
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), V2) -> a__U62(a__isNatIList(V2))
  , a__U62(X) -> U62(X)
  , a__U62(tt()) -> tt()
  , a__U71(X1, X2, X3) -> U71(X1, X2, X3)
  , a__U71(tt(), L, N) -> a__U72(a__isNat(N), L)
  , a__U72(X1, X2) -> U72(X1, X2)
  , a__U72(tt(), L) -> s(a__length(mark(L)))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNat(V1))
  , a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) -> a__U71(a__isNatList(L), L, N)
  , a__length(nil()) -> 0()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(0()) -> 0()
  , mark(zeros()) -> a__zeros()
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(nil()) -> nil()
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(length(X)) -> a__length(mark(X))
  , mark(U11(X)) -> a__U11(mark(X))
  , mark(U21(X)) -> a__U21(mark(X))
  , mark(U31(X)) -> a__U31(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U42(X)) -> a__U42(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X)) -> a__U52(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , mark(U62(X)) -> a__U62(mark(X))
  , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
  , mark(U72(X1, X2)) -> a__U72(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U81(X)) -> a__U81(mark(X))
  , mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4)
  , mark(U92(X1, X2, X3, X4)) -> a__U92(mark(X1), X2, X3, X4)
  , mark(U93(X1, X2, X3, X4)) -> a__U93(mark(X1), X2, X3, X4)
  , a__U81(X) -> U81(X)
  , a__U81(tt()) -> nil()
  , a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4)
  , a__U91(tt(), IL, M, N) -> a__U92(a__isNat(M), IL, M, N)
  , a__U92(X1, X2, X3, X4) -> U92(X1, X2, X3, X4)
  , a__U92(tt(), IL, M, N) -> a__U93(a__isNat(N), IL, M, N)
  , a__U93(X1, X2, X3, X4) -> U93(X1, X2, X3, X4)
  , a__U93(tt(), IL, M, N) -> cons(mark(N), take(M, IL))
  , a__take(X1, X2) -> take(X1, X2)
  , a__take(0(), IL) -> a__U81(a__isNatIList(IL))
  , a__take(s(M), cons(N, IL)) ->
    a__U91(a__isNatIList(IL), IL, M, N) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {14} by applications of
Pre({14}) =
{11,13,15,16,17,18,19,20,21,22,24,25,27,28,29,30,32,33,34,35,40}.
Here rules are labeled as follows:

  DPs:
    { 1: a__U41^#(tt(), V2) ->
         c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
    , 2: a__isNatIList^#(V) ->
         c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
    , 3: a__isNatIList^#(cons(V1, V2)) ->
         c_15(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 4: a__isNatList^#(cons(V1, V2)) ->
         c_22(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 5: a__isNatList^#(take(V1, V2)) ->
         c_24(a__U61^#(a__isNat(V1), V2), a__isNat^#(V1))
    , 6: a__isNat^#(s(V1)) ->
         c_35(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
    , 7: a__isNat^#(length(V1)) ->
         c_36(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
    , 8: a__U51^#(tt(), V2) ->
         c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
    , 9: a__U61^#(tt(), V2) ->
         c_26(a__U62^#(a__isNatIList(V2)), a__isNatIList^#(V2))
    , 10: a__U71^#(tt(), L, N) ->
          c_30(a__U72^#(a__isNat(N), L), a__isNat^#(N))
    , 11: a__U72^#(tt(), L) -> c_32(a__length^#(mark(L)), mark^#(L))
    , 12: a__length^#(cons(N, L)) ->
          c_38(a__U71^#(a__isNatList(L), L, N), a__isNatList^#(L))
    , 13: mark^#(cons(X1, X2)) -> c_40(mark^#(X1))
    , 14: mark^#(zeros()) -> c_42(a__zeros^#())
    , 15: mark^#(s(X)) -> c_44(mark^#(X))
    , 16: mark^#(take(X1, X2)) ->
          c_46(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
    , 17: mark^#(length(X)) -> c_47(a__length^#(mark(X)), mark^#(X))
    , 18: mark^#(U11(X)) -> c_48(a__U11^#(mark(X)), mark^#(X))
    , 19: mark^#(U21(X)) -> c_49(a__U21^#(mark(X)), mark^#(X))
    , 20: mark^#(U31(X)) -> c_50(a__U31^#(mark(X)), mark^#(X))
    , 21: mark^#(U41(X1, X2)) ->
          c_51(a__U41^#(mark(X1), X2), mark^#(X1))
    , 22: mark^#(U42(X)) -> c_52(a__U42^#(mark(X)), mark^#(X))
    , 23: mark^#(isNatIList(X)) -> c_53(a__isNatIList^#(X))
    , 24: mark^#(U51(X1, X2)) ->
          c_54(a__U51^#(mark(X1), X2), mark^#(X1))
    , 25: mark^#(U52(X)) -> c_55(a__U52^#(mark(X)), mark^#(X))
    , 26: mark^#(isNatList(X)) -> c_56(a__isNatList^#(X))
    , 27: mark^#(U61(X1, X2)) ->
          c_57(a__U61^#(mark(X1), X2), mark^#(X1))
    , 28: mark^#(U62(X)) -> c_58(a__U62^#(mark(X)), mark^#(X))
    , 29: mark^#(U71(X1, X2, X3)) ->
          c_59(a__U71^#(mark(X1), X2, X3), mark^#(X1))
    , 30: mark^#(U72(X1, X2)) ->
          c_60(a__U72^#(mark(X1), X2), mark^#(X1))
    , 31: mark^#(isNat(X)) -> c_61(a__isNat^#(X))
    , 32: mark^#(U81(X)) -> c_62(a__U81^#(mark(X)), mark^#(X))
    , 33: mark^#(U91(X1, X2, X3, X4)) ->
          c_63(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1))
    , 34: mark^#(U92(X1, X2, X3, X4)) ->
          c_64(a__U92^#(mark(X1), X2, X3, X4), mark^#(X1))
    , 35: mark^#(U93(X1, X2, X3, X4)) ->
          c_65(a__U93^#(mark(X1), X2, X3, X4), mark^#(X1))
    , 36: a__take^#(0(), IL) ->
          c_75(a__U81^#(a__isNatIList(IL)), a__isNatIList^#(IL))
    , 37: a__take^#(s(M), cons(N, IL)) ->
          c_76(a__U91^#(a__isNatIList(IL), IL, M, N), a__isNatIList^#(IL))
    , 38: a__U91^#(tt(), IL, M, N) ->
          c_69(a__U92^#(a__isNat(M), IL, M, N), a__isNat^#(M))
    , 39: a__U92^#(tt(), IL, M, N) ->
          c_71(a__U93^#(a__isNat(N), IL, M, N), a__isNat^#(N))
    , 40: a__U93^#(tt(), IL, M, N) -> c_73(mark^#(N))
    , 41: a__zeros^#() -> c_1()
    , 42: a__zeros^#() -> c_2()
    , 43: a__U11^#(X) -> c_3()
    , 44: a__U11^#(tt()) -> c_4()
    , 45: a__U21^#(X) -> c_5()
    , 46: a__U21^#(tt()) -> c_6()
    , 47: a__U31^#(X) -> c_7()
    , 48: a__U31^#(tt()) -> c_8()
    , 49: a__U41^#(X1, X2) -> c_9()
    , 50: a__U42^#(X) -> c_11()
    , 51: a__U42^#(tt()) -> c_12()
    , 52: a__isNatIList^#(X) -> c_14()
    , 53: a__isNatIList^#(zeros()) -> c_16()
    , 54: a__isNatList^#(X) -> c_21()
    , 55: a__isNatList^#(nil()) -> c_23()
    , 56: a__isNat^#(X) -> c_33()
    , 57: a__isNat^#(0()) -> c_34()
    , 58: a__U51^#(X1, X2) -> c_17()
    , 59: a__U52^#(X) -> c_19()
    , 60: a__U52^#(tt()) -> c_20()
    , 61: a__U61^#(X1, X2) -> c_25()
    , 62: a__U62^#(X) -> c_27()
    , 63: a__U62^#(tt()) -> c_28()
    , 64: a__U71^#(X1, X2, X3) -> c_29()
    , 65: a__U72^#(X1, X2) -> c_31()
    , 66: a__length^#(X) -> c_37()
    , 67: a__length^#(nil()) -> c_39()
    , 68: mark^#(0()) -> c_41()
    , 69: mark^#(tt()) -> c_43()
    , 70: mark^#(nil()) -> c_45()
    , 71: a__take^#(X1, X2) -> c_74()
    , 72: a__U81^#(X) -> c_66()
    , 73: a__U81^#(tt()) -> c_67()
    , 74: a__U91^#(X1, X2, X3, X4) -> c_68()
    , 75: a__U92^#(X1, X2, X3, X4) -> c_70()
    , 76: a__U93^#(X1, X2, X3, X4) -> c_72() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { a__U41^#(tt(), V2) ->
    c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__isNatIList^#(V) ->
    c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
  , a__isNatIList^#(cons(V1, V2)) ->
    c_15(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(cons(V1, V2)) ->
    c_22(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(take(V1, V2)) ->
    c_24(a__U61^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNat^#(s(V1)) -> c_35(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(length(V1)) ->
    c_36(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U51^#(tt(), V2) ->
    c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U61^#(tt(), V2) ->
    c_26(a__U62^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__U71^#(tt(), L, N) ->
    c_30(a__U72^#(a__isNat(N), L), a__isNat^#(N))
  , a__U72^#(tt(), L) -> c_32(a__length^#(mark(L)), mark^#(L))
  , a__length^#(cons(N, L)) ->
    c_38(a__U71^#(a__isNatList(L), L, N), a__isNatList^#(L))
  , mark^#(cons(X1, X2)) -> c_40(mark^#(X1))
  , mark^#(s(X)) -> c_44(mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_46(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(length(X)) -> c_47(a__length^#(mark(X)), mark^#(X))
  , mark^#(U11(X)) -> c_48(a__U11^#(mark(X)), mark^#(X))
  , mark^#(U21(X)) -> c_49(a__U21^#(mark(X)), mark^#(X))
  , mark^#(U31(X)) -> c_50(a__U31^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2)) -> c_51(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U42(X)) -> c_52(a__U42^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_53(a__isNatIList^#(X))
  , mark^#(U51(X1, X2)) -> c_54(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X)) -> c_55(a__U52^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_56(a__isNatList^#(X))
  , mark^#(U61(X1, X2)) -> c_57(a__U61^#(mark(X1), X2), mark^#(X1))
  , mark^#(U62(X)) -> c_58(a__U62^#(mark(X)), mark^#(X))
  , mark^#(U71(X1, X2, X3)) ->
    c_59(a__U71^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U72(X1, X2)) -> c_60(a__U72^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_61(a__isNat^#(X))
  , mark^#(U81(X)) -> c_62(a__U81^#(mark(X)), mark^#(X))
  , mark^#(U91(X1, X2, X3, X4)) ->
    c_63(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U92(X1, X2, X3, X4)) ->
    c_64(a__U92^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U93(X1, X2, X3, X4)) ->
    c_65(a__U93^#(mark(X1), X2, X3, X4), mark^#(X1))
  , a__take^#(0(), IL) ->
    c_75(a__U81^#(a__isNatIList(IL)), a__isNatIList^#(IL))
  , a__take^#(s(M), cons(N, IL)) ->
    c_76(a__U91^#(a__isNatIList(IL), IL, M, N), a__isNatIList^#(IL))
  , a__U91^#(tt(), IL, M, N) ->
    c_69(a__U92^#(a__isNat(M), IL, M, N), a__isNat^#(M))
  , a__U92^#(tt(), IL, M, N) ->
    c_71(a__U93^#(a__isNat(N), IL, M, N), a__isNat^#(N))
  , a__U93^#(tt(), IL, M, N) -> c_73(mark^#(N)) }
Weak DPs:
  { a__zeros^#() -> c_1()
  , a__zeros^#() -> c_2()
  , a__U11^#(X) -> c_3()
  , a__U11^#(tt()) -> c_4()
  , a__U21^#(X) -> c_5()
  , a__U21^#(tt()) -> c_6()
  , a__U31^#(X) -> c_7()
  , a__U31^#(tt()) -> c_8()
  , a__U41^#(X1, X2) -> c_9()
  , a__U42^#(X) -> c_11()
  , a__U42^#(tt()) -> c_12()
  , a__isNatIList^#(X) -> c_14()
  , a__isNatIList^#(zeros()) -> c_16()
  , a__isNatList^#(X) -> c_21()
  , a__isNatList^#(nil()) -> c_23()
  , a__isNat^#(X) -> c_33()
  , a__isNat^#(0()) -> c_34()
  , a__U51^#(X1, X2) -> c_17()
  , a__U52^#(X) -> c_19()
  , a__U52^#(tt()) -> c_20()
  , a__U61^#(X1, X2) -> c_25()
  , a__U62^#(X) -> c_27()
  , a__U62^#(tt()) -> c_28()
  , a__U71^#(X1, X2, X3) -> c_29()
  , a__U72^#(X1, X2) -> c_31()
  , a__length^#(X) -> c_37()
  , a__length^#(nil()) -> c_39()
  , mark^#(0()) -> c_41()
  , mark^#(zeros()) -> c_42(a__zeros^#())
  , mark^#(tt()) -> c_43()
  , mark^#(nil()) -> c_45()
  , a__take^#(X1, X2) -> c_74()
  , a__U81^#(X) -> c_66()
  , a__U81^#(tt()) -> c_67()
  , a__U91^#(X1, X2, X3, X4) -> c_68()
  , a__U92^#(X1, X2, X3, X4) -> c_70()
  , a__U93^#(X1, X2, X3, X4) -> c_72() }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X) -> U11(X)
  , a__U11(tt()) -> tt()
  , a__U21(X) -> U21(X)
  , a__U21(tt()) -> tt()
  , a__U31(X) -> U31(X)
  , a__U31(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), V2) -> a__U42(a__isNatIList(V2))
  , a__U42(X) -> U42(X)
  , a__U42(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatList(V))
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), V2) -> a__U52(a__isNatList(V2))
  , a__U52(X) -> U52(X)
  , a__U52(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2)
  , a__isNatList(nil()) -> tt()
  , a__isNatList(take(V1, V2)) -> a__U61(a__isNat(V1), V2)
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), V2) -> a__U62(a__isNatIList(V2))
  , a__U62(X) -> U62(X)
  , a__U62(tt()) -> tt()
  , a__U71(X1, X2, X3) -> U71(X1, X2, X3)
  , a__U71(tt(), L, N) -> a__U72(a__isNat(N), L)
  , a__U72(X1, X2) -> U72(X1, X2)
  , a__U72(tt(), L) -> s(a__length(mark(L)))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNat(V1))
  , a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) -> a__U71(a__isNatList(L), L, N)
  , a__length(nil()) -> 0()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(0()) -> 0()
  , mark(zeros()) -> a__zeros()
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(nil()) -> nil()
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(length(X)) -> a__length(mark(X))
  , mark(U11(X)) -> a__U11(mark(X))
  , mark(U21(X)) -> a__U21(mark(X))
  , mark(U31(X)) -> a__U31(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U42(X)) -> a__U42(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X)) -> a__U52(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , mark(U62(X)) -> a__U62(mark(X))
  , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
  , mark(U72(X1, X2)) -> a__U72(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U81(X)) -> a__U81(mark(X))
  , mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4)
  , mark(U92(X1, X2, X3, X4)) -> a__U92(mark(X1), X2, X3, X4)
  , mark(U93(X1, X2, X3, X4)) -> a__U93(mark(X1), X2, X3, X4)
  , a__U81(X) -> U81(X)
  , a__U81(tt()) -> nil()
  , a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4)
  , a__U91(tt(), IL, M, N) -> a__U92(a__isNat(M), IL, M, N)
  , a__U92(X1, X2, X3, X4) -> U92(X1, X2, X3, X4)
  , a__U92(tt(), IL, M, N) -> a__U93(a__isNat(N), IL, M, N)
  , a__U93(X1, X2, X3, X4) -> U93(X1, X2, X3, X4)
  , a__U93(tt(), IL, M, N) -> cons(mark(N), take(M, IL))
  , a__take(X1, X2) -> take(X1, X2)
  , a__take(0(), IL) -> a__U81(a__isNatIList(IL))
  , a__take(s(M), cons(N, IL)) ->
    a__U91(a__isNatIList(IL), IL, M, N) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ a__zeros^#() -> c_1()
, a__zeros^#() -> c_2()
, a__U11^#(X) -> c_3()
, a__U11^#(tt()) -> c_4()
, a__U21^#(X) -> c_5()
, a__U21^#(tt()) -> c_6()
, a__U31^#(X) -> c_7()
, a__U31^#(tt()) -> c_8()
, a__U41^#(X1, X2) -> c_9()
, a__U42^#(X) -> c_11()
, a__U42^#(tt()) -> c_12()
, a__isNatIList^#(X) -> c_14()
, a__isNatIList^#(zeros()) -> c_16()
, a__isNatList^#(X) -> c_21()
, a__isNatList^#(nil()) -> c_23()
, a__isNat^#(X) -> c_33()
, a__isNat^#(0()) -> c_34()
, a__U51^#(X1, X2) -> c_17()
, a__U52^#(X) -> c_19()
, a__U52^#(tt()) -> c_20()
, a__U61^#(X1, X2) -> c_25()
, a__U62^#(X) -> c_27()
, a__U62^#(tt()) -> c_28()
, a__U71^#(X1, X2, X3) -> c_29()
, a__U72^#(X1, X2) -> c_31()
, a__length^#(X) -> c_37()
, a__length^#(nil()) -> c_39()
, mark^#(0()) -> c_41()
, mark^#(zeros()) -> c_42(a__zeros^#())
, mark^#(tt()) -> c_43()
, mark^#(nil()) -> c_45()
, a__take^#(X1, X2) -> c_74()
, a__U81^#(X) -> c_66()
, a__U81^#(tt()) -> c_67()
, a__U91^#(X1, X2, X3, X4) -> c_68()
, a__U92^#(X1, X2, X3, X4) -> c_70()
, a__U93^#(X1, X2, X3, X4) -> c_72() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { a__U41^#(tt(), V2) ->
    c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__isNatIList^#(V) ->
    c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
  , a__isNatIList^#(cons(V1, V2)) ->
    c_15(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(cons(V1, V2)) ->
    c_22(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(take(V1, V2)) ->
    c_24(a__U61^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNat^#(s(V1)) -> c_35(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(length(V1)) ->
    c_36(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U51^#(tt(), V2) ->
    c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U61^#(tt(), V2) ->
    c_26(a__U62^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__U71^#(tt(), L, N) ->
    c_30(a__U72^#(a__isNat(N), L), a__isNat^#(N))
  , a__U72^#(tt(), L) -> c_32(a__length^#(mark(L)), mark^#(L))
  , a__length^#(cons(N, L)) ->
    c_38(a__U71^#(a__isNatList(L), L, N), a__isNatList^#(L))
  , mark^#(cons(X1, X2)) -> c_40(mark^#(X1))
  , mark^#(s(X)) -> c_44(mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_46(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(length(X)) -> c_47(a__length^#(mark(X)), mark^#(X))
  , mark^#(U11(X)) -> c_48(a__U11^#(mark(X)), mark^#(X))
  , mark^#(U21(X)) -> c_49(a__U21^#(mark(X)), mark^#(X))
  , mark^#(U31(X)) -> c_50(a__U31^#(mark(X)), mark^#(X))
  , mark^#(U41(X1, X2)) -> c_51(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U42(X)) -> c_52(a__U42^#(mark(X)), mark^#(X))
  , mark^#(isNatIList(X)) -> c_53(a__isNatIList^#(X))
  , mark^#(U51(X1, X2)) -> c_54(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X)) -> c_55(a__U52^#(mark(X)), mark^#(X))
  , mark^#(isNatList(X)) -> c_56(a__isNatList^#(X))
  , mark^#(U61(X1, X2)) -> c_57(a__U61^#(mark(X1), X2), mark^#(X1))
  , mark^#(U62(X)) -> c_58(a__U62^#(mark(X)), mark^#(X))
  , mark^#(U71(X1, X2, X3)) ->
    c_59(a__U71^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U72(X1, X2)) -> c_60(a__U72^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_61(a__isNat^#(X))
  , mark^#(U81(X)) -> c_62(a__U81^#(mark(X)), mark^#(X))
  , mark^#(U91(X1, X2, X3, X4)) ->
    c_63(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U92(X1, X2, X3, X4)) ->
    c_64(a__U92^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U93(X1, X2, X3, X4)) ->
    c_65(a__U93^#(mark(X1), X2, X3, X4), mark^#(X1))
  , a__take^#(0(), IL) ->
    c_75(a__U81^#(a__isNatIList(IL)), a__isNatIList^#(IL))
  , a__take^#(s(M), cons(N, IL)) ->
    c_76(a__U91^#(a__isNatIList(IL), IL, M, N), a__isNatIList^#(IL))
  , a__U91^#(tt(), IL, M, N) ->
    c_69(a__U92^#(a__isNat(M), IL, M, N), a__isNat^#(M))
  , a__U92^#(tt(), IL, M, N) ->
    c_71(a__U93^#(a__isNat(N), IL, M, N), a__isNat^#(N))
  , a__U93^#(tt(), IL, M, N) -> c_73(mark^#(N)) }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X) -> U11(X)
  , a__U11(tt()) -> tt()
  , a__U21(X) -> U21(X)
  , a__U21(tt()) -> tt()
  , a__U31(X) -> U31(X)
  , a__U31(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), V2) -> a__U42(a__isNatIList(V2))
  , a__U42(X) -> U42(X)
  , a__U42(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatList(V))
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), V2) -> a__U52(a__isNatList(V2))
  , a__U52(X) -> U52(X)
  , a__U52(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2)
  , a__isNatList(nil()) -> tt()
  , a__isNatList(take(V1, V2)) -> a__U61(a__isNat(V1), V2)
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), V2) -> a__U62(a__isNatIList(V2))
  , a__U62(X) -> U62(X)
  , a__U62(tt()) -> tt()
  , a__U71(X1, X2, X3) -> U71(X1, X2, X3)
  , a__U71(tt(), L, N) -> a__U72(a__isNat(N), L)
  , a__U72(X1, X2) -> U72(X1, X2)
  , a__U72(tt(), L) -> s(a__length(mark(L)))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNat(V1))
  , a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) -> a__U71(a__isNatList(L), L, N)
  , a__length(nil()) -> 0()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(0()) -> 0()
  , mark(zeros()) -> a__zeros()
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(nil()) -> nil()
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(length(X)) -> a__length(mark(X))
  , mark(U11(X)) -> a__U11(mark(X))
  , mark(U21(X)) -> a__U21(mark(X))
  , mark(U31(X)) -> a__U31(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U42(X)) -> a__U42(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X)) -> a__U52(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , mark(U62(X)) -> a__U62(mark(X))
  , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
  , mark(U72(X1, X2)) -> a__U72(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U81(X)) -> a__U81(mark(X))
  , mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4)
  , mark(U92(X1, X2, X3, X4)) -> a__U92(mark(X1), X2, X3, X4)
  , mark(U93(X1, X2, X3, X4)) -> a__U93(mark(X1), X2, X3, X4)
  , a__U81(X) -> U81(X)
  , a__U81(tt()) -> nil()
  , a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4)
  , a__U91(tt(), IL, M, N) -> a__U92(a__isNat(M), IL, M, N)
  , a__U92(X1, X2, X3, X4) -> U92(X1, X2, X3, X4)
  , a__U92(tt(), IL, M, N) -> a__U93(a__isNat(N), IL, M, N)
  , a__U93(X1, X2, X3, X4) -> U93(X1, X2, X3, X4)
  , a__U93(tt(), IL, M, N) -> cons(mark(N), take(M, IL))
  , a__take(X1, X2) -> take(X1, X2)
  , a__take(0(), IL) -> a__U81(a__isNatIList(IL))
  , a__take(s(M), cons(N, IL)) ->
    a__U91(a__isNatIList(IL), IL, M, N) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { a__U41^#(tt(), V2) ->
    c_10(a__U42^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , a__isNatIList^#(V) ->
    c_13(a__U31^#(a__isNatList(V)), a__isNatList^#(V))
  , a__isNat^#(s(V1)) -> c_35(a__U21^#(a__isNat(V1)), a__isNat^#(V1))
  , a__isNat^#(length(V1)) ->
    c_36(a__U11^#(a__isNatList(V1)), a__isNatList^#(V1))
  , a__U51^#(tt(), V2) ->
    c_18(a__U52^#(a__isNatList(V2)), a__isNatList^#(V2))
  , a__U61^#(tt(), V2) ->
    c_26(a__U62^#(a__isNatIList(V2)), a__isNatIList^#(V2))
  , mark^#(U11(X)) -> c_48(a__U11^#(mark(X)), mark^#(X))
  , mark^#(U21(X)) -> c_49(a__U21^#(mark(X)), mark^#(X))
  , mark^#(U31(X)) -> c_50(a__U31^#(mark(X)), mark^#(X))
  , mark^#(U42(X)) -> c_52(a__U42^#(mark(X)), mark^#(X))
  , mark^#(U52(X)) -> c_55(a__U52^#(mark(X)), mark^#(X))
  , mark^#(U62(X)) -> c_58(a__U62^#(mark(X)), mark^#(X))
  , mark^#(U81(X)) -> c_62(a__U81^#(mark(X)), mark^#(X))
  , a__take^#(0(), IL) ->
    c_75(a__U81^#(a__isNatIList(IL)), a__isNatIList^#(IL)) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { a__U41^#(tt(), V2) -> c_1(a__isNatIList^#(V2))
  , a__isNatIList^#(V) -> c_2(a__isNatList^#(V))
  , a__isNatIList^#(cons(V1, V2)) ->
    c_3(a__U41^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(cons(V1, V2)) ->
    c_4(a__U51^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNatList^#(take(V1, V2)) ->
    c_5(a__U61^#(a__isNat(V1), V2), a__isNat^#(V1))
  , a__isNat^#(s(V1)) -> c_6(a__isNat^#(V1))
  , a__isNat^#(length(V1)) -> c_7(a__isNatList^#(V1))
  , a__U51^#(tt(), V2) -> c_8(a__isNatList^#(V2))
  , a__U61^#(tt(), V2) -> c_9(a__isNatIList^#(V2))
  , a__U71^#(tt(), L, N) ->
    c_10(a__U72^#(a__isNat(N), L), a__isNat^#(N))
  , a__U72^#(tt(), L) -> c_11(a__length^#(mark(L)), mark^#(L))
  , a__length^#(cons(N, L)) ->
    c_12(a__U71^#(a__isNatList(L), L, N), a__isNatList^#(L))
  , mark^#(cons(X1, X2)) -> c_13(mark^#(X1))
  , mark^#(s(X)) -> c_14(mark^#(X))
  , mark^#(take(X1, X2)) ->
    c_15(a__take^#(mark(X1), mark(X2)), mark^#(X1), mark^#(X2))
  , mark^#(length(X)) -> c_16(a__length^#(mark(X)), mark^#(X))
  , mark^#(U11(X)) -> c_17(mark^#(X))
  , mark^#(U21(X)) -> c_18(mark^#(X))
  , mark^#(U31(X)) -> c_19(mark^#(X))
  , mark^#(U41(X1, X2)) -> c_20(a__U41^#(mark(X1), X2), mark^#(X1))
  , mark^#(U42(X)) -> c_21(mark^#(X))
  , mark^#(isNatIList(X)) -> c_22(a__isNatIList^#(X))
  , mark^#(U51(X1, X2)) -> c_23(a__U51^#(mark(X1), X2), mark^#(X1))
  , mark^#(U52(X)) -> c_24(mark^#(X))
  , mark^#(isNatList(X)) -> c_25(a__isNatList^#(X))
  , mark^#(U61(X1, X2)) -> c_26(a__U61^#(mark(X1), X2), mark^#(X1))
  , mark^#(U62(X)) -> c_27(mark^#(X))
  , mark^#(U71(X1, X2, X3)) ->
    c_28(a__U71^#(mark(X1), X2, X3), mark^#(X1))
  , mark^#(U72(X1, X2)) -> c_29(a__U72^#(mark(X1), X2), mark^#(X1))
  , mark^#(isNat(X)) -> c_30(a__isNat^#(X))
  , mark^#(U81(X)) -> c_31(mark^#(X))
  , mark^#(U91(X1, X2, X3, X4)) ->
    c_32(a__U91^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U92(X1, X2, X3, X4)) ->
    c_33(a__U92^#(mark(X1), X2, X3, X4), mark^#(X1))
  , mark^#(U93(X1, X2, X3, X4)) ->
    c_34(a__U93^#(mark(X1), X2, X3, X4), mark^#(X1))
  , a__take^#(0(), IL) -> c_35(a__isNatIList^#(IL))
  , a__take^#(s(M), cons(N, IL)) ->
    c_36(a__U91^#(a__isNatIList(IL), IL, M, N), a__isNatIList^#(IL))
  , a__U91^#(tt(), IL, M, N) ->
    c_37(a__U92^#(a__isNat(M), IL, M, N), a__isNat^#(M))
  , a__U92^#(tt(), IL, M, N) ->
    c_38(a__U93^#(a__isNat(N), IL, M, N), a__isNat^#(N))
  , a__U93^#(tt(), IL, M, N) -> c_39(mark^#(N)) }
Weak Trs:
  { a__zeros() -> cons(0(), zeros())
  , a__zeros() -> zeros()
  , a__U11(X) -> U11(X)
  , a__U11(tt()) -> tt()
  , a__U21(X) -> U21(X)
  , a__U21(tt()) -> tt()
  , a__U31(X) -> U31(X)
  , a__U31(tt()) -> tt()
  , a__U41(X1, X2) -> U41(X1, X2)
  , a__U41(tt(), V2) -> a__U42(a__isNatIList(V2))
  , a__U42(X) -> U42(X)
  , a__U42(tt()) -> tt()
  , a__isNatIList(V) -> a__U31(a__isNatList(V))
  , a__isNatIList(X) -> isNatIList(X)
  , a__isNatIList(cons(V1, V2)) -> a__U41(a__isNat(V1), V2)
  , a__isNatIList(zeros()) -> tt()
  , a__U51(X1, X2) -> U51(X1, X2)
  , a__U51(tt(), V2) -> a__U52(a__isNatList(V2))
  , a__U52(X) -> U52(X)
  , a__U52(tt()) -> tt()
  , a__isNatList(X) -> isNatList(X)
  , a__isNatList(cons(V1, V2)) -> a__U51(a__isNat(V1), V2)
  , a__isNatList(nil()) -> tt()
  , a__isNatList(take(V1, V2)) -> a__U61(a__isNat(V1), V2)
  , a__U61(X1, X2) -> U61(X1, X2)
  , a__U61(tt(), V2) -> a__U62(a__isNatIList(V2))
  , a__U62(X) -> U62(X)
  , a__U62(tt()) -> tt()
  , a__U71(X1, X2, X3) -> U71(X1, X2, X3)
  , a__U71(tt(), L, N) -> a__U72(a__isNat(N), L)
  , a__U72(X1, X2) -> U72(X1, X2)
  , a__U72(tt(), L) -> s(a__length(mark(L)))
  , a__isNat(X) -> isNat(X)
  , a__isNat(0()) -> tt()
  , a__isNat(s(V1)) -> a__U21(a__isNat(V1))
  , a__isNat(length(V1)) -> a__U11(a__isNatList(V1))
  , a__length(X) -> length(X)
  , a__length(cons(N, L)) -> a__U71(a__isNatList(L), L, N)
  , a__length(nil()) -> 0()
  , mark(cons(X1, X2)) -> cons(mark(X1), X2)
  , mark(0()) -> 0()
  , mark(zeros()) -> a__zeros()
  , mark(tt()) -> tt()
  , mark(s(X)) -> s(mark(X))
  , mark(nil()) -> nil()
  , mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
  , mark(length(X)) -> a__length(mark(X))
  , mark(U11(X)) -> a__U11(mark(X))
  , mark(U21(X)) -> a__U21(mark(X))
  , mark(U31(X)) -> a__U31(mark(X))
  , mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
  , mark(U42(X)) -> a__U42(mark(X))
  , mark(isNatIList(X)) -> a__isNatIList(X)
  , mark(U51(X1, X2)) -> a__U51(mark(X1), X2)
  , mark(U52(X)) -> a__U52(mark(X))
  , mark(isNatList(X)) -> a__isNatList(X)
  , mark(U61(X1, X2)) -> a__U61(mark(X1), X2)
  , mark(U62(X)) -> a__U62(mark(X))
  , mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
  , mark(U72(X1, X2)) -> a__U72(mark(X1), X2)
  , mark(isNat(X)) -> a__isNat(X)
  , mark(U81(X)) -> a__U81(mark(X))
  , mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4)
  , mark(U92(X1, X2, X3, X4)) -> a__U92(mark(X1), X2, X3, X4)
  , mark(U93(X1, X2, X3, X4)) -> a__U93(mark(X1), X2, X3, X4)
  , a__U81(X) -> U81(X)
  , a__U81(tt()) -> nil()
  , a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4)
  , a__U91(tt(), IL, M, N) -> a__U92(a__isNat(M), IL, M, N)
  , a__U92(X1, X2, X3, X4) -> U92(X1, X2, X3, X4)
  , a__U92(tt(), IL, M, N) -> a__U93(a__isNat(N), IL, M, N)
  , a__U93(X1, X2, X3, X4) -> U93(X1, X2, X3, X4)
  , a__U93(tt(), IL, M, N) -> cons(mark(N), take(M, IL))
  , a__take(X1, X2) -> take(X1, X2)
  , a__take(0(), IL) -> a__U81(a__isNatIList(IL))
  , a__take(s(M), cons(N, IL)) ->
    a__U91(a__isNatIList(IL), IL, M, N) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..